Subspaces with normalized tight frame wavelets in \(\mathbb{R}\) (Q2781335)

Let \(\psi \in L^2(\mathbb{R})\). A normalized tight frame wavelet for a subspace \(X\subset L^2(\mathbb{R})\) is a collection of functions \(\{\psi_{j,k}(x) = 2^{j/2} \psi(2^jx-k)\), \(j \in Z\), \(k \in Z\}\), with the property NEWLINE\[NEWLINE\forall f \in X, \quad f = \sum \langle f, \psi_{j,k} \rangle \psi_{j,k}.NEWLINE\]NEWLINE A measurable set \(E \subset\mathbb{R}\) is called a normalized tight frame wavelet set for \(X\) if the function \(\psi\) is of the form \(\hat \psi = 1/(2\pi) \chi_E\). In a similar manner one can define tight frame wavelet sets, frame wavelet sets, and Bessel sets. \(X\) is called a reducing subspace of \(L^2(\mathbb{R})\) if it is invariant under dilations by 2 and translations by integers. NEWLINENEWLINENEWLINEIn this paper the authors study the properties of the Bessel sets and normalized tight frame wavelet sets. In particular, they provide characterizations of the latter for reducing subspaces. It is also shown that reducing subspaces are exactly the spaces \(X\) that admit a Bessel set which is a normalized tight frame wavelet set for \(X\). Some of these results were generalized to higher dimensions in a subsequent work of \textit{X. Dai, Y. Diao, Q. Gu}, and \textit{D. Han} [Proc. Am. Math. Soc. 130, No. 11, 3259-3267 (2002; Zbl 1004.42025, following review)].